WASHINGTON—The Trump administration is backing a $13 billion increase in funding for the World Bank, putting aside its skepticism of the big government-backed institutions that manage the global economy, in part because it wants the World Bank as a counterweight to China’s growing international influence.

The change, which will allow the bank to increase lending to poor-country clients, comes after what European and other officials described as difficult negotiations over tough terms demanded by the U.S. One official described...